These are the kind of social, medical, psychological reasons I have experience of/have heard of:
wheelchair bound child appealing to get into the only local school with disability access: this had to be combined with medical reports showing why same child could not be bussed long distances to go into less popular but accessible school at other end of LEA
child with longterm psychological problems who needed to go to a school where she already knew someone: evidence consisted of letters from CAHMS (mentalh health services) and GP
child whose mother was terminally ill allowed into only local school which offered bereavement counselling
child who had been severely bullied (psychological stress documented) allowed to get into school away from bullies: note that this is not a passport into one particular school, just away from one particular school
Note that this is a two-stage process: you need to show not only why School A will not do, but (far more importantly) why School B is the only one that will: otherwise the LEA will want to send her to the unpopular and undersubscribed School C at the other end of the boundaries, rather than have to overstep their numbers
and any argument that teaching standards at School C are substandard are not likely to go down well at all
so you'd need to find something that is specific to School B (your choice) and be able to motivate why your child needs this more than other children: after all, there is no reason for the LEA to think that some other child deserves the scrotty School C more than yours does; and there are so many of these top 10% children: if the top 10% in every other school in the LEA insisted of getting into this popular school- well, it would get impossible; so LEA would be afraid of creating a precedent
We were the parents of the wheelchair bound child. We still had to take it to appeal.